We apply small cancellation methods originating from group theory to investigate the structure of a quotient ring $\mathbb{ Z}_2\mathcal{ F} / \mathcal{ I}$, where $\mathbb{ Z}_2\mathcal{ F}$ is the group algebra of the free group $\mathcal{ F}$ over the field $\mathbb{ Z}_2$, and the ideal $\mathcal{ I}$ is generated by a single trinomial $1 + v + vw$, where $v$ is a complicated word depending on $w$. In $\mathbb{ Z}_2\mathcal{ F} / \mathcal{ I}$ we have $(1 + w)^{ -1} = v$, so $1 + w$ becomes invertible. We construct an explicit linear basis of $\mathbb{ Z}_2\mathcal{ F} / \mathcal{ I}$ (thus showing that $\mathbb{ Z}_2\mathcal{ F} / \mathcal{ I}\neq 0$). This is the first step in constructing rings with exotic properties.
Categories: math.RA